How can I determine the interest rate, given increasing monthly deposits and a final amount?
I need help to determine the interest rate of this investment.
The initial value is nothing
The monthly investment is R500.
It stays R500 a month until the year is over and increases by R50. Thus, R550 per month the 2nd year, and R600 the 3rd year, etc.
Total period is 10 Years
Total invested: R87000 over the 10 years.
Total returned (deposits plus interest): R149028
can someone who knows the formula please help.
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I'm assuming that compound interest is paid monthly. In that case, the interest rate is ~0.9162% per month (i.e. an APR of 11.57%).
The following formula is the one you are interested in:
sum_{y=0}^9 (sum_{m=0}^11 (500+50*y)*(1+T)^(120-12*y-m))=149028
I didn't attempt to solve this, but instead used trial and error. The value T=0.0091624 works nicely; you can verify this in Wolfram Alpha.
Getting at APR/EAR is simply (1+T)^12 - 1.
A method to work this out can be found by using a simpler quarterly example over two years. Using an example rate, r = 0.01, this is the example calculation for the first year
y1q1 = 0 + 500
y1q2 = y1q1 (1 + r) + 500
y1q3 = y1q2 (1 + r) + 500
y1q4 = y1q3 (1 + r) + 500
y1q4 (1 + r) = 2050.502505
Equivalent to the summation
Continuing, this is the calculation for two years
y2q1 = y1q4 (1 + r) + 500 + 50
y2q2 = y2q1 (1 + r) + 500 + 50
y2q3 = y2q2 (1 + r) + 500 + 50
y2q4 = y2q3 (1 + r) + 500 + 50
y2q4 (1 + r) = 4389.313885
Equivalent to this summation
To create a general formula this needs to be re-expressed as a double summation, where n is the total number of periods, n = 8
This can be generalised, where
y is the number of years
m is the number of months or quarters (or days)
p is the initial regular deposit
d is the annual deposit increase
By induction, this can be reduced to a formula
Checking
r = 0.01
p = 500
d = 50
y = 2
m = 4
n = 8
((1 + r)^(1 + n) (d + p (-1 + (1 + r)^m) +
(1 + r)^(-m y) (-d + p + d y -
(1 + r)^m (p + d y))))/(r (-1 + (1 + r)^m)) = 4389.313885
This can be used to solve for the OP's values
fv = 149028
p = 500
d = 50
y = 10
m = 12
n = 120
Plot of future value for a range of r also showing the target fv
Solving exactly yields r = 0.009162396432
Giving an annual effective rate of
(1 + 0.009162396432)^12 - 1 = 0.115662 = 11.5662 %
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